Gravity SeriesSide quest · Optional
Free fall and elapsed time
Clocks taking different routes between the same two events can record different amounts of time. Comparing those readings gives another way to understand free fall.
This optional article compares clocks travelling along different paths. It offers another way to describe free fall. Continue to How spatial geometry affects light for the next main lesson.
Taking a clock on a journey
A clock’s motion affects how much time it records. Imagine a spacecraft carrying a clock past a row of synchronised clocks floating at rest relative to one another. Compare their readings as it passes. The spacecraft’s clock accumulates less time than the row of clocks records for the same journey.
At everyday speeds, the difference is tiny. At about 86.6% of light speed, the travelling clock records one second for every two seconds measured by the row of clocks. The time recorded by a clock along its own journey is called proper time.
Two clocks between the same two events
Imagine throwing a ball straight up and catching it at the same height two seconds later. Attach a tiny clock to it, and keep another beside your hand.
Both clocks start together and meet again at the catch. Their readings differ slightly because one stayed at the same height while the other travelled up and back.
Height and speed both matter
The travelling clock spends some time higher up, where clocks run faster. It also moves relative to the ground, which reduces the time it records. Both effects contribute to the reading when it returns.
We can compare other journeys too. A clock could be carried a little higher, or follow a lower arc, provided it still returns to the same place at exactly the same moment. Those alternative journeys may need an engine or some other force to produce them.
For the short flight shown here, the freely falling arc records more time than nearby alternative paths. With no air resistance and gravity treated as constant, a launch speed of about 9.8 metres per second gives a two-second flight and a maximum height of about 4.9 metres.
The ball does not need to choose
This can sound as though the ball selects the journey that gives its clock the most time. The comparison is something we calculate; the ball only follows the motion set by its initial velocity and the surrounding spacetime.
Describing the path through its clock reading gives the same answer as working out the motion step by step. It is another way of expressing the free-fall rule.
How far the rule applies
The maximum-time result applies to a short enough section of a free-fall path, compared with nearby alternatives between the same events. It is not a guarantee that every long journey through curved spacetime gives the greatest possible clock reading.
The more general mathematical rule uses the word stationary. It means that the elapsed-time result has a locally flat slope as we vary the path, as a graph does at the top of a smooth hill. In this example, that point is a maximum.